Spatial dispersion of crystals as a critical problem for deep UV lithography A. G. Serebryakov,* F. Bociort, and J. Braat
Delft Technical University, Delft, the Netherlands ͑Submitted February 14, 2003͒ ͑ ͒ Opticheski Zhurnal 70,45–49 August 2003 The birefringence effect caused by spatial dispersion in crystals has been known for more than a hundred years, but until recently it was studied only from the viewpoint of theoretical physics. This article discusses the mathematical description of the effect as well as problems associated with the presence of the effect in modern lithographic optical systems and methods of solving these problems. © 2003 Optical Society of America
BIREFRINGENCE CAUSED BY SPATIAL DISPERSION IN circumstances. In practice, the presence of this effect means LITHOGRAPHY that pairs of polarized rays can appear after refraction at each surface, with the refractive index for each of the rays de- The birefringence effect caused by spatial dispersion pending on the wavelength and propagation direction of the ͑SD͒ was discovered theoretically in 1877 by Lorentz.1 In incident ray. the 1960s, Agranovich and Ginzburg gave a theoretical de- In general, the displacement vector D and the electric scription of the effect, based on Maxwell’s equations.2 Quan- field vector E are mutually dependent, following from Max- titative measurements for semiconductors were first made in 3 well’s equations, where the inverse dielectric tensor 1972. Ϫ1 The possibility that the SD effect would influence the ij ( ,k) is determined by frequency and the direction of image quality in lithography was a surprise for the lithogra- wave vector k: phy industry, since SD can be a critical problem for the cre- ͑ ͒ϭ Ϫ1͑ ͒ ͑ ͒ ͑ ͒ ation of a new generation of lithography machines with a Ei ,k ͚ ij ,k D j ,k , 1 wavelength of 157 nm and lens elements made from crystal- j 4–6 line materials. The world’s leading producers of microcir- c2 c2 cuits initiated practical measurements of birefringence for ϭϪ ͓ ͑ " ͔͒ϭϪ ͓ ͑ ͑Ϫ1 ͔͒͒ ͑ ͒ D 2 k k E 2 k k• D . 2 CaF2 , the most widely used crystal in lithography, carried out by the National Institute of Standards and Technology Since the wave vector is kϭksϭ(˜n/c)(,s)s, where c is ͑ ͒ 7–9 ⌬ ϭ USA . The value obtained for 157 nm was n (11.8 the speed of light in vacuum and s is the unit vector along the Ϯ ϫ Ϫ7 ͑ ⌬ 0.4) 10 where n is the difference between the re- propagation direction of the ray, we have fractive indices for the ordinary and extraordinary rays͒. The most critical consequences of SD in lithography are the pres- 1 s͓s ͑Ϫ1D͔͒ϩ Dϭ0, ͑3͒ ence of quasi-interference effects in the image plane of the • n2 optical system, caused by polarization of the rays, and dou- bling of the rays when they are refracted at each surface. where n is the refractive index. Losses of image contrast in the optical system, caused by the Since vector D is orthogonal to the wave vector and presence of these effects, can be more than 50% of the initial s"Dϭ0, it is possible to choose a coordinate system XYZ so value, depending on the optical layout. The producers of that D lies in the XY plane and vector s is parallel to the Z lithographic machines were already acquainted with birefrin- axis: gence caused by residual stresses in crystals after polishing. However, the SD value is several times as large as the pre- 0 sϭͩ 0 ͪ . ͑4͒ viously known dispersion effect caused by residual stresses, and new optical layouts and solutions consequently need to 1 be developed to compensate it. Equation ͑3͒ for the XY plane can be simplified by in- Ϫ1 Ϫ1 troducing the transverse component Ќ of tensor . Then SPATIAL DISPERSION IN CRYSTALS the eigenvalue equation with eigenvectors DЈ and DЉ and eigenvalue 1/n2 can be written in general as The essence of the SD effect is that the permittivity depends not only on the wavelength but also on the direction Ϫ 1 1Dϭ D. ͑5͒ in which the wave propagates in the medium. The effect Ќ n2 must be taken into account in media with regular structure ͑crystals͒ if the wavelength of the radiation becomes compa- For a single component of vector D, rable with the dimensions of the crystal lattice. The effect thus needs to allowed for in the far UV when using even Ϫ 1 ␦ D ϭͩ 1͑,k͒Ϫ ␦ ͪ D , ͑6͒ those crystals that are considered isotropic under ordinary ij j ijЌ n2 ij j
566 J. Opt. Technol. 70 (8), August 2003 1070-9762/2003/080566-04$20.00 © 2003 The Optical Society of America 566 ␦ ␦ ϭ ϭ ␦ where ij is the Kronecker delta ( ij 1ifi j, and ij ϭ Þ Ϫ1 0ifi j). Since tensor ij ( ,k) is real and symmetrical, the eigenvalues are real and the eigenvectors are orthogonal. Consequently, in an anisotropic medium, the polarization states of DЈ and DЉ are determined by the two eigenvectors. Thus, the physical properties of an anisotropic material lead to the existence of two mutually orthogonal plane- polarized waves that propagate through a medium character- Ϫ1 ized by tensor ij ( ,k). To determine the refractive index of the crystal, it is necessary to find the eigenvalues of the Ϫ1 projection of tensor ij in a plane orthogonal to k. The refractive indices for the two polarized rays are then deter- ϭͱ Ϫ1 mined from the eigenvalues; i.e., n 1/eigval( ijЌ). Since the relative value of the effect is small, a series expansion in FIG. 1. Coordinate system of the crystal lattice. powers of the wave-vector components can be used to deter- Ϫ1 ϭϪ1 mine the inverse dielectric tensor: ij ( ,k) ij ( ) ϩ ␦ ϩ Ϫ1 i ijl( )kl ijlm( )klkm , where ij ( ) describes the the Z axis coincides with the optic axis, while the X and Y ␦ ordinary frequency dispersion, tensor ijl( ) characterizes axes are orthogonal to it. In our case, there is also a coordi-  the gyrotropy, and tensor ijlm( ) shows how tensor nate system of the crystal lattice. This can be a Cartesian Ϫ1 ij ( ,k) depends on the SD. coordinate system coinciding with the edges of the CaF  2 In general, tensor ijlm( ) has 81 components, but, be- cubic lattice or a system of spherical coordinates in which cause of the high degree of symmetry of the CaF2 crystal the ray intersects the lattice in a direction determined by ͑ 5 ͒ symmetry type Oh or m3m , the number of independent radius vector r, coinciding with unit wave vector s ͑Fig. 1͒.    components reduces to three: 1111 , 1122 , and 1212 . More- In this case, according to the rules adopted in crystallogra- ␦ ϭ over, in our case, ijl( ) 0. It can be shown that the phy, the directions of the coordinate axes are denoted as amount of birefringence, i.e., the difference between the two ͓100͔, ͓010͔, and ͓001͔, the face diagonals as ͓110͔, ͓101͔, refractive indices for the two polarization states, is determin- and ͓011͔, and, finally, the cube diagonal as ͓111͔. ined from Ϫ1 In general, since we have a tensor ij ( ,k) with known 1 eigenvalues and eigenvectors and a measured birefringence ⌬n͑,k͒ϭ n͑͒3͑ Ϫ Ϫ2 ͒ for a certain definite direction like ͓100͔ or ͓111͔,wecan 2 1111 1122 1212 rotate this definite tensor to an arbitrary propagation direc- ϫ 2͑ ͑␦ 2͒ Ϫ ͑␦ 2͒ ͒ ͑ ͒ k eigval2 ijli Ќ eigval1 ijli Ќ , 8 tion of the light wave. In a spherical coordinate system, we have an orthogonal set of vectors ͑, , r͒, and, to diagonal- where l1 , l2 , and l3 are the direction cosines of the angles ize a tensor in the plane of transverse k, it is necessary to find between vector k and the coordinate axes. The refractive- the projection of the tensor in the two-dimensional space of index variations can be determined by a single measurement and , of the birefringence and by computing the eigenvalues for ͑␦ 2͒ ϭ ͑ ͑ ͒␦ 2 Ϫ1͑ ͒͒ ͑ ͒ each propagation direction. eigval1,2 ijli Ќ eigval1,2 R , ijli R , , 9 where R(,) and RϪ1(,) are rotation matrices. Then the SPATIAL DEPENDENCE OF THE BIREFRINGENCE eigenvalues of the tensor determine the refractive indices, To describe the spatial dependence of the birefringence and, to determine the polarization states, it is necessary to for an arbitrary propagation direction, it is necessary to find the inverse projections of the eigenvectors in three- choose a coordinate system for the crystal lattice. A Carte- dimensional space. The value of the birefringence is deter- sian coordinate system is ordinarily used in optics in which mined from
1 1 1 ⌬ ϳͩ ͑␦ 2͒ Ϫ ͑␦ 2͒ ϭ 2 ͱ 4 ͑ ϩ ͒2ϩ 4 ϩ 2 2 ͑ Ϫ ͒ͪ n eigval l Ќ eigval l Ќ sin cos 7 cos 4 sin 2 cos sin 2 cos 4 5 . 2 ij i 1 ij i 16 4 4 ͑10͒
To describe the SD effect in a cubic crystal, coordinates quently, a cubic crystal can be called heptaxial with respect and are varied from 0 to 2. Then in a solid angle of to SD. In all, there are seven axes along which the effect 360° there are twelve maxima and fourteen minima of the equals zero: three axes coinciding with the cube edges ͕100͖, birefringence ͑Fig. 2͒. The figure shows the refractive-index and four cube diagonals ͕111͖. However, these two sets of difference of various directions in a cubic crystal. Conse- axes have different refractive indices. The maximum of the
567 J. Opt. Technol. 70 (8), August 2003 Serebryakov et al. 567 FIG. 3. Angular dependence of the SD.
FIG. 2. Birefringence maxima and minima in a CaF crystal. 2 COMPENSATING THE SD EFFECT IN LITHOGRAPHY The results of computing the birefringence for three di- effect is observed in the ͕110͖ directions ͑the face diagonals͒. rections are shown in Table I, where the character of the The most interesting behavior of the dependence of the bire- dependence of the SD over the pupil of the optical system is fringence is that in the plane of the cube diagonal, i.e., for shown for an axial point of the image. For an off-axis point, ϭ/4, ϭ0to/2 ͑Fig. 3͒. some distortion of the radial pattern of the effect is observed.
TABLE I. Spatial dependence of birefringence.
568 J. Opt. Technol. 70 (8), August 2003 Serebryakov et al. 568 To maintain the symmetry of the optical system, only A possible solution of the problem can be to create a three directions in the crystal—along ͕100͖, ͕110͖, and ‘‘mixed’’ crystal. The SD effect is also present in other crys- ͕ ͖ 111 —can be used for the optic axis, with the dependence of tals with cubic symmetry, SrF2 and BaF2 , but has the oppo- the effect on the angle of incidence being more symmetrical site sign. Thus, it is theoretically possible to create a crystal ͕ ͖ ͕ ͖ ͕ ͖ in the 100 and 110 directions than along 111 , where with the formula Ca1ϪxSrxF2 or Ca1ϪxBaxF2 , in which SD there is a change in the sign and the character of the variation for 157 nm is absent. However, in this case, the solution of of the effect. However, the ͕111͖ direction is preferable from the problem involves crystal synthesis. the viewpoint of processing ͑polishing͒ the crystals, since the residual stresses are minimal for this direction. In principle, a CONCLUSION ͕ ͖ random direction abc can be taken as the optic axis, but This article has explained the main problems that arise in the computation of the effect and the technological problems modern lithography in connection with the appearance of the can be unpredictable in this case. spatial dispersion effect. A mathematical description of the The general strategy of compensating the effect is as effect and the possibilities of compensating it have been pro- follows: It can be seen from the dependence displayed in Fig. posed. 3 that the effect can be mutually compensated, for example, The authors thank the ASML Corp. ͑the Netherlands͒ for by choosing the orientation of the components. Moreover, support of the research and Dr. Burnett from the National ͕ ͖ ͕ ͖ for the 100 and 111 directions, the effect is absent for rays Institute of Standards and Technology ͑USA͒ for valuable ͕ ͖ close to the optic axis. The 100 direction has a symmetry of advice when preparing the article. rotation of 45°, while ͕111͖ has a symmetry of 60°. Thus, it is possible to adjust the individual elements to each other, *Email: [email protected] with the formation of a wavefront with minimum distortions. The minimum number of components for such a compensa- 1 H. A. Lorentz, Collected Papers II, III ͑Martinus Nijhoff, The Hague, ͕ ͖ ͕ ͖ ͕ ͖ 1936͒. tion method is four (0° for 100 , 45° for 100 , 0° for 111 , 2 ͕ ͖͒ V. M. Agronovich and V. L. Ginzburg, Crystal Optics with Spatial Dis- and 45° for 111 . The possibilities of compensation for an persion, and Excitons ͑Nauka, Moscow, 1979; Springer-Verlag, Berlin, off-axis point are still unclear because of the complex char- 1984͒. acter of the dependence of the effect. 3 P. Y. Yu and M. Cardona, in Computational Solid-State Physics, edited by ͑ ͒ Difficulties in designing such optical systems are also F. Herman et al. Plenum Press, New York, 1972 ,p.7. 4 M. Becchi, C. Oldano, and S. Ponti, ‘‘Spatial dispersion and optics of associated with the rate of calculating the ray path in the crystals,’’ J. Optics A 1, 713 ͑1999͒. crystal, since, for each ray at each interface of the air–crystal 5 J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, ‘‘Aberrations medium, it is necessary to compute two refractive indices by introduced by a lens made from a birefringent material,’’ Appl. Opt. 39, 592 ͑2000͒. the technique indicated above. Of the software available on 6 Y. Unno, ‘‘Influence of birefringence on the image formation of high- the market, only one program ͑Code V, ©Optical Research resolution projection optics,’’ Appl. Opt. 39, 3243 ͑2000͒. Associates͒ includes a procedure for calculating the ray path 7 J. H. Burnett, Z. H. Levine, and E. L. Shirley, ‘‘Intrinsic birefringence in taking SD into account. However, the doubling of the ray at calcium fluoride and barium fluoride,’’ Phys. Rev. B 64, 241 102 ͑2000͒. 8 J. H. Burnett, Z. H. Levine, E. L. Shirley, and J. H. Bruning, ‘‘Symmetry each surface is ignored, whereas, even with twenty surfaces of spatial-dispersion-induced birefringence and its implications for CaF 20 6 2 in the system, there can be 2 Ϸ10 rays in the image plane ultraviolet optics,’’ J. Microlith. Microfab. Microsys. 1, 213 ͑2002͒. for each object point. 9 http://physics.nist.gov/Divisions/Div842/Gp3/DUVMatChar/birefring.html
569 J. Opt. Technol. 70 (8), August 2003 Serebryakov et al. 569